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# LineContainer

Authors: Benjamin Qi, Andi Qu

### Prerequisites

Convex Containers

## Half-Plane Intersection

Resources
CF
PetrExpected linear time!

### This section is not complete.

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Example problem + implementation?
StatusSourceProblem NameDifficultyTagsSolution
KattisNormalView Solution
JOIHardView Solution
Balkan OIVery Hard
Show Tags

Geometry, Binary Search

External Sol

## LineContainer (aka $O(N \log N)$ CHT)

StatusSourceProblem NameDifficultyTagsSolution
YSNormalView Solution
Resources
KACTLsource of code that I (Ben) use
cp-algorelated topic (but not the same)

### Example Problem

Focus Problem – read through this problem before continuing!

#### Analysis

Instead of focusing on the pillars that should be destroyed, let's instead focus on the pillars that remain.

The total cost consists of the cost due to height differences plus the cost of destroying unused pillars. The latter cost is equal to the cost to destroy all pillars minus the cost to destroy the remaining pillars.

Since the cost to destroy all pillars is constant, we can thus turn the problem into one about building pillars instead of destroying them!

From this, we get a basic DP recurrence. Let $dp[i]$ be the minimum cost to build the bridge so that the last build pillar is pillar $i$.

$dp = -w_1$ and the following recurrence holds:

\begin{aligned} dp[i] &= \min_{j < i}(dp[j] + (h_j - h_i)^2 - w_i)\\ &= \min_{j < i}(dp[j] + h_j^2 - 2h_ih_j) + h_i^2 - w_i \end{aligned}

Notice how

$dp[j] + h_j^2 - 2h_ih_j$

effectively describes a linear function $y = mx + c$, where $m = -2h_j$, $x = h_i$, and $c = dp[j] + h_j^2$

This means that we can use CHT to compute $dp[i]$ efficiently!

However, since $m$ is not monotonic, we can't use linear CHT using a deque, so we must settle with $O(N \log N)$.

Code

## Problems

StatusSourceProblem NameDifficultyTagsSolution
YSNormalView Solution
POINormal
Show Tags

DP, convex

View Solution
CEOIHard
Show Tags

DP, convex

External Sol
FHCHard
Show Tags

DP, convex

Check FHC
ACHardCheck AC
TLXHardCheck TLX
Old GoldHardExternal Sol